U.S. application Ser. No. 14/364,060, which published as US 2014/0309730 and is incorporated herein by reference in its entirety, describes implanting a tissue engaging member having a cinching loop that runs through the tissue engaging member on a mitral or tricuspid valve annulus, waiting for a tissue healing process to form a bond between the tissue engaging member and the annulus, and subsequently cinching the cinching loop to reduce the diameter of the annulus.